Now we explain the ideas of the proof. Let's firstly recall Kazaras-Khuri-Lee's result in [1]}, where they proved the Gromov-Hausdorff convergence under additional conditions that \(H_2(M,\mathbb {Z})=0\)  and Ricci curvature bounded uniformly from below by \(\text{Ric}_g \ge -2\Lambda \) . Their basic ingredient is the mass inequality proved by Bray-Kazaras-Khuri-Stern in [2]}, which is the following (see Section REF  for more details)
\(m(g) \ge \frac{1}{16\pi }\int _{M_{ext}}\left( \frac{|\nabla ^2u^j|^2}{|\nabla u^j|}+R_g|\nabla u^j| \right) dV,\) 
